Page:Popular Science Monthly Volume 70.djvu/87

chosen, I should so make this choice that these movements carry the finger $D$ to the point originally occupied by the finger $D'$, that is, to the point $M$; this finger $D$ will thus be in contact with the object $a$, which will make it feel the impression $A$.

I then make the movements corresponding to the series $\sigma$; in these movements, by hypothesis, the position of the finger $D$ does not change, this finger therefore remains in contact with the object $a$ and continues to feel the impression $A$. Finally I make the movements corresponding to the series $S'$. As $S'$ is inverse to $S$, these movements carry the finger $D'$ to the point previously occupied by the finger $D$, that is, to the point $M$. If, as may be supposed, the object a has not budged, this finger $D'$ will be in contact with this object and will feel anew the impression $A'$. . . . $Q.E.D.$

Let us see the consequences. I consider a series of muscular sensations $\Sigma$. To this series will correspond a point $M$ of the first tactile space. Now take again the two series $s$ and $s'$, inverses of one another, of which we have just spoken. To the series $s+\Sigma +s'$ will correspond a point $N$ of the second tactile space, since to any series of muscular sensations corresponds, as we have said, a point, whether in the first space or in the second.

I am going to consider the two points $N$ and $M$, thus defined, as corresponding. What authorizes me so to do? For this correspondence to be admissible, it is necessary that if two points $M$ and $M'$, corresponding in the first space to two series $\Sigma$ and $\Sigma '$, are identical, so also are the two corresponding points of the second space $N$ and $N'$, that is the two points which correspond to the two series $s+\Sigma +s'$ and $s+\Sigma '+s'$. Now we shall see that this condition is fulfilled.

First a remark. As $S$ and $S'$ are inverses of one another, we shall have $S+S'=0$, and consequently $S+S'+\Sigma =\Sigma +S+S'=\Sigma$, or again $\Sigma +S+S'+\Sigma '=\Sigma +\Sigma '$; but it does not follow that we have $S+\Sigma +S'=\Sigma$; because, though we have used the addition sign to represent the succession of our sensations, it is clear that the order of this succession is not indifferent: we can not, therefore, as in ordinary addition, invert the order of the terms; to use abridged language, our operations are associative, but not commutative.

That fixed, in order that $\Sigma$ and $\Sigma '$ should correspond to the same point $M=M'$ of the first space, it is necessary and sufficient for us to have $\Sigma '=\Sigma +\sigma$. We shall then have: $S+\Sigma '+\Sigma '=S+\Sigma +\sigma +S'=8+\Sigma +S'+S+\sigma +S'$.

But we have just ascertained that $S+\sigma +8'$ was one of the series $\sigma$. We shall therefore have: $S+\Sigma '+S'=S+\Sigma +S'+\sigma '$, which means that the series $S+\Sigma '+S'$ and $S+\Sigma +S'$ correspond to the same point $N=N'$ of the second space. $Q.E.D.$

Our two spaces therefore correspond point for point; they can be